AMC 12 2023b Problem 12

For complex numbers $u=a+bi$ and $v=c+di$, define the binary operation $\otimes$ by$$u\otimes v=ac+bdi.$$Suppose $z$ is a complex number such that $z\otimes z=z^{2}+40$. What is $|z|$?

$\textbf{(A) }2 \textbf{(B) }5 \textbf{(C) }\sqrt{5} \textbf{(D) }\sqrt{10} \textbf{(E) }5\sqrt{2}$